Problem: William works out for $\frac{4}{5}$ of an hour every day. To keep his exercise routines interesting, he includes different types of exercises, such as sit-ups and squats, in each workout. If each type of exercise takes $\frac{4}{15}$ of an hour, how many different types of exercise can William do in each workout?
Answer: To find out how many types of exercise William could do in each workout, divide the total amount of exercise time ( $\frac{4}{5}$ of an hour) by the amount of time each exercise type takes ( $\frac{4}{15}$ of an hour). $ \dfrac{{\dfrac{4}{5} \text{ hour}}} {{\dfrac{4}{15} \text{ hour per exercise}}} = {\text{ number of exercises}} $ Dividing by a fraction is the same as multiplying by the reciprocal. The reciprocal of ${\dfrac{4}{15} \text{ hour per exercise}}$ is ${\dfrac{15}{4} \text{ exercises per hour}}$ $ {\dfrac{4}{5}\text{ hour}} \times {\dfrac{15}{4} \text{ exercises per hour}} = {\text{ number of exercises}} $ $ \dfrac{{4} \cdot {15}} {{5} \cdot {4}} = {\text{ number of exercises}} $ Reduce terms with common factors by dividing the $4$ in the numerator and the $4$ in the denominator by $4$ $ \dfrac{{\cancel{4}^{1}} \cdot {15}} {{5} \cdot {\cancel{4}^{1}}} = {\text{ number of exercises}} $ Reduce terms with common factors by dividing the $15$ in the numerator and the $5$ in the denominator by $5$ $ \dfrac{{1} \cdot {\cancel{15}^{3}}} {{\cancel{5}^{1}} \cdot {1}} = {\text{ number of exercises}} $ Simplify: $ \dfrac{{1} \cdot {3}} {{1} \cdot {1}} = {3} $ William can do 3 different types of exercise per workout.